The flow of $X$ is the map $\psi_{s,t}=\phi_s\circ\phi_t^{-1}:m\in M\mapsto\phi_s(\phi_t^{-1}(m))\in M$.
For a tensor field $T$ on $M$ its Lie derivative along the time dependent vector field $X_t$ is given by: $$\frac{d}{dt}(\psi_{t,s}^\ast T)=\psi_{s,t}^\ast(\mathcal{L}(X_t).T).$$

Concerning this expression I suppose that in general, in this context, $\mathcal{L}(X_t).(\psi_{s,t}^\ast T)\neq\psi_{s,t}^\ast(\mathcal{L}(X_t).T)$, but I don't know an example exhibiting such inequality.
Probably it is silly but my question is: where I can look for such an example?